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  3. Soliton Equations and their Algebro-Geometric Solutions
  • Cited by 181
  • Volume 1: (1+1)-Dimensional Continuous Models
  • Fritz Gesztesy, University of Missouri, Columbia, Helge Holden, Norwegian University of Science and Technology, Trondheim
Publisher:
Cambridge University Press
Online publication date:
December 2009
Print publication year:
2003
Online ISBN:
9780511546723
DOI:
https://doi.org/10.1017/CBO9780511546723
Subjects:
Mathematics (general), Mathematics, Differential and Integral Equations, Dynamical Systems and Control Theory, Mathematical Physics
Series:
Cambridge Studies in Advanced Mathematics (79)
Information

Book description

The focus of this book is on algebro-geometric solutions of completely integrable nonlinear partial differential equations in (1+1)-dimensions, also known as soliton equations. Explicitly treated integrable models include the KdV, AKNS, sine-Gordon, and Camassa-Holm hierarchies as well as the classical massive Thirring system. An extensive treatment of the class of algebro-geometric solutions in the stationary as well as time-dependent contexts is provided. The formalism presented includes trace formulas, Dubrovin-type initial value problems, Baker-Akhiezer functions, and theta function representations of all relevant quantities involved. The book uses techniques from the theory of differential equations, spectral analysis, and elements of algebraic geometry (most notably, the theory of compact Riemann surfaces). The presentation is rigorous, detailed, and self-contained, with ample background material provided in various appendices. Detailed notes for each chapter together with an exhaustive bibliography enhance the presentation offered in the main text.

Reviews

'… this is a book that I would recommend to any student of mine, for clarity and completeness of exposition … Any expert as well would enjoy the book and learn something stimulating from the sidenotes that point to alternative developments. We look forward to volumes two and three!'

Source: Mathematical Reviews

'The book is very well organized and carefully written. It could be particularly useful for analysts wanting to learn new methods coming from algebraic geometry.'

Source: EMS Newsletter

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